标题： 小波和图$C^*$-代数 显示英文标题

标题： Wavelets and graph $C^*$-algebras

摘要： 在这里，我们概述了小波理论与图$C^{\ast}$-代数表示论之间的联系，包括 A. Kumjian 和 D. Pask 的高秩图$C^*$-代数。在过去20年里，许多作者研究了这种联系的不同方面，我们首先在这篇论文中回顾了已知的结果。 然后，我们讨论了几种新的方法来推广这些结果，并获得与高秩图表示相关的波let。在\cite{FGKP}中，我们介绍了与高秩图相关的“立方体波let”。在这里，我们将这种构造推广，以构建任意形状的波let。我们还提出了另一种与高秩图相关的波let构造，我们预计这将有助于网络上的流量分析。最后，我们将\cite{hammond}的谱图波let推广到高秩图，给出了与高秩图相关的第三类波let。 显示英文摘要

摘要： Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.